Geometry of Non-Linear Continuum Mechanics

This is an attempt to contribute to bring the discussion on non-linear issues in Continuum Mechanics (CM) under the comprehensive supervision of modern differential geometry (DG). Some people would say that engineers do not need so much mathematics, but such an opinion is a sure indicator of lack of knowledge of the matter. In fact I am an engineer, a University teacher and a designer in Structural Engineering and my experience is that a lot of troubles would have been saved to me by a basic training in DG.

Anyway, it has been just a scant adoption of notions and concepts of DG which have contributed to waste of time and energies in debates about ill-posed questions. As a matter of fact, long-lasting discussions and difficulties on fundamental and computational issues can be resolved by a natural, but carefull, geometric path of reasoning.

A first contribution to this line of approach to CM has been published in the paper

Thank you for the interesting discussion. I disagree with your reasoning regarding F not being an acceptable deformation measure.

I agree that the Lie derivative of F along the motion vanishes (I don't think this is "unfortunate"; it is simply a fact). It is true that you can choose different connections (in the ambient space here for a covariant time derivative). Let us not forget that we do need to choose a metric in the ambient space (and reference configuration) to be able to make a scalar out of F (a two-point tensor). This may seem "unnatural" or unpleasant but in elasticity we need to know how distances are measured in the ambient space. If an elastic sheet is attached to a curved 2-manifold, obviously, one must have the ambient space metric to calculate the stored energy. The point is: you must explicitly include the metrics of the ambient space and reference configuration explicitly in your stored energy (or Lagrangian density). Now having a metric the natural connection (in the ambient space) would be the Levi-Civita connection. However, you could choose any other connection and you would end up with a different measure of "time derivative" of F. This doesn't make the theory weak or inconsistent in any sense nor does it make F an unacceptable measure of deformation. Is it a disaster that one can have different measures of stress in continuum mechanics? No. As long as the appropriate deformation measure is chosen the theory is consistent.

Another point: if you choose F as a deformation measure then in calculating the Lie derivative of stored energy (assuming you're looking at nonlinear elasticity) Lie derivative of F vanishes but the Lie derivative of g (ambient space metric) does not and it would be your "rate of strain". If you choose to take a covariant time derivative you will have the covariant time derivative of F but the covariant time derivative of g vanishes (assuming the Levi-Civita connection is used).

Now regarding "lower dimensional bodies" the deformed body is a submanifold of the ambient space manifold and so you can use the induced connection to get the right covariant time derivative.

I agree that multiplicative decomposition of F has some issues but it is not a "failure". I also agree with you calling the "intermediate configuration", "intermediate linear space".

let me say that it is a pleasure to have an opportunity to discuss about fundamental issues with a scientist and researcher in Continuum Mechanics (CM) with a firm background in Differential Geometry (DG).

I would like to bringto your attention the following issues.

1) The metric tensor field in the space (and in the time) affine manifolds are basic ingredients of the theory of CM. Different choices lead to different theories.

2) The space-time manifold endowed with a metric tensor field is a Riemann manifold and the Levi-Civita connection is the only torsion-free (i.e. symmetric) and metric connection in that manifold, but different connections may and are in fact often considered, for instance the ones induced by general coordinate systems, which may well be non-symmetric. Connection dependent definitions of time-rates are then not natural.

3) Given a connection in the ambient space manifold, the induced connection in a placement of a lower dimensional body (a submanifold of the ambient space manifold) defines a covariant derivative which at each point is performed along vectors tangent to the submanifold. Covariant time derivatives are then outside its domain because they act along the velocity vector whichmay not be tangent to the submanifold.

To develop a general theory of CM the most suitable DG context is provided by the four-dimensional events manifold and by the trajectory submanifold.

1) I agree that metric tensor is a fundamental ingredient of any geometric theory of continuum mechanics. In the case of solids, for the reference manifold one may lose the freedom in choosing an arbitrary metric if the reference configuration is required to be stress-free.

2) I agree with your statement regarding the Levi-Civita connection. Here, perhaps you should clarify what you precisely mean by "natural". The Levi-Civita connection is not always the natural connection. For example, given a solid with distributed dislocations the natural connection (in the reference configuration) is a Weitzenbock connection (metric-compatible, flat, but with torsion). This was first realized by Kondo and Bilby in the 1950s.

3) The induced connection acts in the tangent space of the submanifold. I don't think this is the main issue in this discussion though.

The ambient space is a Riemannian manifold. So, if you need a time derivative for F, covariant time derivative using the Levi-Civita connection is perfectly fine, in my opinion. If you think this is not a natural time derivative, I would like to know what your definition of a "natural time derivative" is.

Thank you Arash and Givoanni for taking the time to explain your points of view. I don't understand much of it andmy understanding may be totally wrong.

Gioovanni's point appears to be that if there is no unique connection that applies to solid mechanics, then we are in the same situation as when nounique objective rates was the rage.Giovanni, could you explain your natural connection.

Arash says that general (local?) covariance is all we need; which is esentially objectivity in another guise. Arash, could you explain what covariance phyically means?

Also, since we are not dealing with relativistic mechanics, aren't things simpler in solid mechanics? Can we just take results from relativity and run with them?

How can these theories be guided and given a solid foundation by experiment? Is anything predicted that's unusual which can be used to provide us with a preference for the right definition of derivative?

come on gents, let's not get carried away and say that \dot F is not defined precsiely in continuum mechanics and a 'right' definition is required.

Just go take a look at Gurtin's Introduction to Continuum Mechanics. It is competely precise and self-consistent.

Now you might want to get all worked up and say I do not want to consider the tangent space of two points of a configuration of 3-d body immersed in 3-d Euclidean point space as the same - that is your choice, but that does not mean that the deformation gradient rate cannot be given precise mathematical (and physical) meaning within the context of differentiation in finite dimensional vector spaces. You recognize a point space and its associated translation space and learn a little about Frechet Derivatives and off you go.

Let's not make heavy weather of the things that we have some control over and know how to make precise sense of. There is bigger fish to fry in continuum mechanics - like solving some hard nonlinear PDE that arise in continuum mechanics - I can assure you no amount of differential geometrizing will ease that burden.

I agree with Amit. My point was: Lie derivative of F vanishes ok but the covariant time derivative of F is a well-defined and valid time derivative that is also "objective".

In my opinion, we should avoid what happened to "rational mechanics" people (obsession with notation and funny things like the "principle of equipresence", etc.). If all differential geometry can offer is to make things look more complicated, then there is no real contribution and we should forget about it. Geometry can indeed lead to important contributions that are more than just simple reinterpretations of what is already known. What one should focus on is to solve problems that cannot be solved using the existing theories/approaches.

Let me explain in greater detail the concept of naturality and the geometric motivation why the seemengly appropriate covariant time-derivative in the ambient space is not apt to define the time rate of material tensors (the ones entering in a constitutive relation) which act on vectors tangent to the current body placement.

1) The convective time-derivative, depending only on the motion,
which is an essential ingredient of the theory, is a natural notion.
On the contrary, the choice of a connection in the trajectory manifold to perform
the parallel time-derivative of a material tensor makes naturality of the notion of time-rate to be lost.

2) For lower dimensional bodies
(such as wires and membranes in the 3D Euclid space, sketched in fig. 1,2
the pull-back along the motion transforms material vectors into material vectors
(red arrows in fig. 1,2).
On the contrary, the parallel transport, of immersed material tangent vectors along a particle,
will in general yield a spatial vector which is not tangent to the body placement
(black arrows in fig. 1,2).

Then, for lower dimensional bodies, the definition of time-derivative of F by means of a parallel transport is not acceptable if the result is to be adopted in a constitutive relation because only material vectors (tangent to the body placement) and not spatial vectors (tangent to the ambient space) are admissible.

I am assuming that notions and definition of Continuum Mechanics should be equally be applicable to bodies of any dimensionality, viz. cable, membranes and balls.

As a last observation, I would point out that the theory ensuing from the adoption of the natural definition of time rates is definitely simpler and more suitable for computational applications.

Another point to think about is that Euler's formula for the stretching is an expression of the convective time-derivative of the metric tensor. In accord with duality, the time derivative of the stress is also a convective time-derivative. Why then conceive to take a time derivative by parallel transport when dealing with the deformation gradient? Just because the convective derivative vanishes, but this is an indication that an essntial difficulty is involved. In fact there is no need to consider the time derivative of F in a constitutive relation, the proper candidate to measure deformation being the metric tensor in the ambient space (or better its pull-back to the body placement, to include lower dimensional bodies.

I understand that you like to work with the pulled-back metric and its time derivative. This is perfectly fine. You can equivalently work with the Lie derivative of the spatial metric. I assume you agree that continuum mechanics can be consistently formulated using both spatial and material descriptions? I also understand that you would like to work with a minimal geometric structure. Having the motion Lie derivative can be defined without even any need for a metric. Now my point is: this argument would not be enough for dismissal of F as a deformation measure. Let me emphasize that I am not an advocate for the multiplicative decomposition of deformation gradient but at the same time don't think it is inconsistent. What is problematic is to work with an intermediate "configuration", which is defined only locally and as you mentioned earlier it is not even a configuration.

I also assume that you agree that metric is an essential ingredient and we must have a way of measuring distances to calculate the elastic energy? Then the natural connection in this Riemannian manifold is the Levi-Civita connection (in the classical framework. If one is interested in relativistic effects perhaps more complicated spacetimes can be used to account for cosmological defects) and using the covariant time derivative of F should lead to an acceptable theory. One may still argue what is more natural or fundamental. What I think is an invariant is the time derivative of energy density (a scalar) and this scalar can be written in terms of the time derivative of the pulled-back metric and the second Piola-Kirchhoff stress or the covariant time derivative of F and the first Piola-Kirchhoff stress. What do you think?

Let us also remember that for writing equilibrium equations we do need to use a covariant derivative, i.e. using the Levi-Civita connection is inevitable; divergence of Cauchy stress requires the use of the Levi-Civita connection. I would like to know your opinion on this point too.

I see that a fish fryer came into the blog in a somewhat impolite way.

I don't want to get him carried away but rather to suggest him to learn the art of cooking before spitting his recipes. Wrongly fried fishes may have an awful taste and be dangerous for the health.

For other friends reading this blog, I notice that Fréchet derivatives are defined in linear spaces with a suitable topology. In the discussion opened with this blog the issue is rather how to reconduct a time-derivative along the non-linear trajectory manifold to a (Fréchet) derivation on a linear space. What I sustain is that, among the two possibilities offered by differential geometry (DG), viz. parallel transport and push-pull transformation along the motion, only the second is natural and suitable for material tensor fields entering in constitutive relations. This point was not detected by the treatment by Morton Gurtin in his nice book on Introduction to Continuum Mechanics.

Leaving this discussion apart and looking at DG as a useless fashion, would be as trying to make derivatives with the sole tools of linear algebra.

I am really surprised that also knowledgeable researchers could express agreement with such a stubbornly blind viewpoint. I am not interested in making proselytes with any effort, but rather to discuss in a polite way about scientific issues with other open minded people.

A look at Hill's Invariance in Solid Mechanics (1978) article for when it is convenient to look at objective rate constitutive equations transformed to be in terms of deformation gradient rate and the nominal stress (rate) may be useful here. This turns out to be very useful in the analysis of bvp and bifurcations in the rate-independent setting. This is also all in the context of mechanics of 3-d bodies. Of course, Hill was not a proponent of the multiplicative decomposition and I agree with that viewpoint. I think that what is most important is to understand the transformations between different equivalent representations.

In that article, Hill also explains convected derivatives (in what amounts to a slightly more general setting than with respect to a flow of a manifold - e.g. the Jaumann rate (for a deformable body) can only be understood in that context), that may also be useful for readers of this blog. In my opinion, he does a good job of explaining the Lie derivative idea of pull-back-time derivative-push forward, for a first exposure to the idea for people who do not have any exposure to DG. Once one understands that, then it is perhaps easy to adapt it to the situation when tangent spaces of the same material point along a flow cannot be declared as identical (as for shells, membrane, rods) - of course, it is also in this context that one can appreciate why this is a useful concept.

Notions of DG are of course important in CM for lower-dimensional bodies - people who have done 3-d CM as well as played with shell theories (and implemented them numerically) know and appreciate this. But one of the main questions in this blog had to do with the failure of the multiplicative decomposition, and whether Norris's statement made sense. These developments were made in the context of 3-d bodies in 3-d space - there they make perfect precise sense (at least to me).

There is also a past history of DGizing in CM starting from the works of Simo. It is probably also an open-minded endeavor for readers of this blog (especially young researchers) to take a look at that body of work and come to their own conclusions as to what was learned from that exercise. May be Prof. Romano's paper already covers this point.

People (even the most distinguished) have been confused for a long time with objective time derivatives. I think the main reason has been the lack of familiarity with the geometric techniques. For some reason workers in continuum mechanics have isolated themselves (not all but most) thinking that they are working with a very special theory not realizing that this is just one of the many field theories of physics. Geometry has been essential in many developments in physics and we should learn from that experience. I think you would agree with me?

Deformation gradient has a fixed leg and a leg that moves with the deformable body (two-point tensor). For this reason one cannot take a (naive) time derivative unless everything is Euclidean. Most likely this is what Gurtin does too?

I believe geometric notions are always important when you are dealing with bodies with residual stresses regardless of dimensionality.

Simo was among the very few who both understood geometry and properly used it in computational mechanics. Unfortunately, he died too young.

Let me repeat that I strongly believe solving specific (nonlinear) problems is the right direction in convincing others of usefulness of geometric techniques.

You ask an interesting philosophical question (geometry has been essential in many developments....) that I would enjoy discussing, as we find time. But perhaps you should first start a separate thread.

let me say some words about the
hope that this blog could serve to provide a well-sustained discussion about
the foundations of Continuum Mechanics (CM).
This is the aim why I opened the blog and, in my opinion, the goal can be
reached only if there will be no hurry to claim things just as anyone has
learned in his training, but rather an attitude to think and ponder with
patience and deepness will be experimented, to provide constructive criticisms.

I have learned this lesson once
more at the end of my academic career when I realized that, to overcome a deep
feeling of unsatisfaction for most treatment of CM,

especially concerning rate formulations and constitutive theory,
and to find a rationale out of a highly confused situation, I had to learn fundamental
notions and concepts of Differential Geometry which were not teached to me in
my training as a student of engineering and were later considered in the
scientific community as a mathematical sophisticated discipline of no concern
in applied mechanics.

There is a common old saying in
Naples, Italy, whose translation could be:

The best deaf is who doesn’t intend to listen.

I understand that changing ideas
on fundamentals is a very hard task for experts, but the extraordinary
experience with Einstein’s
relativity theory (absit iniuria verbis)
should learn us to be open minded.

The fortune of that theory was
that people like Hilbert, Klein, Poincaré and Minkowski,
very knowledgeable and authoritative experts, were intrigued by the approach by Einstein and
shared his childish enthusiasm.

There are some significant
similarities between that situation and the present one concerning Continuum
Mechanics.

In both cases one is faced with a
well-consolidated theory rich of implications and interpretations of
experimental facts, but with some unexplicable difficulties and paradoxes.

In both cases the right idea
comes by collecting hints and partial answers by older proposals made by other
valuable researchers.

In the field of our concern,
Continuum Mechanics, the leading new ideas are the following.

2)The statement of the Covariance Paradigm (CP) expressing that the rule for the comparison
between material tensors is the push according to the relevant diffeomorphic
transformation.

The first item is basic but simple and evident, although
never explicitly stated in a proper geometric form. The second item is made
more subtle by a dimensional coincidence between the body and the ambient
space, but is self-proposing for lower dimensional bodies.

To grasp the motivation it should be observed that
comparison between material tensors means that two placements of a body along a
trajectory (whether real or virtual) are considered, as related by the
evolution diffeomorphism between them. The naturalway to make the comparison is just to resort to the evolution
diffeomorphism itself, and to its tangent map, to perform the push-pull
transformations. This idea goes back as far as to Euler and to hiscelebratedformula for thestretching.

The conceptual clarity of this approach is not questionable
and the extraordinary effectiveness of its adoption becomes evident as soon as
it is applied to formulate constitutive relations, to discuss basic issues such
as time independence, time invariance, frame invariance and integrability
conditions.

Fictitious difficulties faced with in CM are swept away by the CP
which leads to formulate rate constitutive relations for
elasto-visco-plasticity (and similar models of material behavior) in a direct
and definite way and resolves the long lasting debate about rates of material
tensors by giving a unique, simple and well-defined answer.

I completely agree that one should be open minded and willing to question what is already accepted by most people. I also believe that in a scientific discussion one should be very specific and clear in asking/answering questions. Let me repeat one of my previous (unanswered) questions. Do you need a connection in writing the balance of linear momentum? If not, how would you write it?

I deeply appreciate your question which will lead us to discuss about a basic issue in Classical Continuum Dynamics (CCD).

I posed the very same question to me some time ago and, in giving a satisfactory answer, a hard effort of investigation was required to me to recover the essential informations from the literature and to extract the underlying geometrical concepts and results, together with new contributions.

I will try to summarize hereafter the basic ideas, focusing on the role of a linear connection in the ambient space and quoting some recent articles by me and co-workers, where the foundations of CCD are treated in detail.

The foundations of CCD are laid down in the most general way by means of a variational principle concerning the trajectory and the relevant evolution operator. The principle may be put in the standard geometrical form of an Action Principle in which the tests in the variational principle are made by displacing the trajectory in the container manifold. To this end the trajectory is considered as a submanifold of the state-space manifold defined to be the velocity-time (or the covelocity-time) manifold and an action one-form on the state-space is devised by a suitable lifting of the scalar lagrangian. Under usual regularity assumptions, the Action Principle may be localized to provide the differential Euler-Lagrangeequation and the corner conditions at singular points.

No geometric connection in the state-space manifold enters into the theory until this point and hence it can be affirmed that CCD may be founded in a natural way in terms of the evolution operator and of the lagrangian, without any additional assumptions, as illustrated in

An equivalent principle can be formulated by imposing that the displacement of the trajectory leaves the energy functional invariant, to get a generalized form of Maupertuis Least Action Principle as illustrated in

The introduction of a linear connection provides however a very valuable tool of investigation about the properties of the evolution along the trajectory fulfilling the Action Principle, the choice of a special connection being a question of convenience. For instance, a curvilinear coordinate system will induce an associated distant parallelism and a corresponding linear connection which has vanishing torsion and curvature forms. The adoption of a Levi-Civita connection will instead induce a torsion-free and metric connection with a non-vanishing curvature. In this respect we must observe that only the torsion of the linear connection enters in the equations of Dynamics. An important example is provided by Poincaré's law of Dynamics which is the outcome of taking the distant parallel transport induced by a mobile reference system associated with curvilinear coordinates. In this case the torsion form is equal to the opposite of the Lie bracket and hence the components of the Lie brackets of the basis vector fields (named structure coefficients) appear into the equation of Dynamics.

The Levi-Civita connection on the trajectory, with the Lagrangian given by the kinetic energy per unit mass, leads to generalized Euler's and d'Alembert's laws of Dynamics. The standard formulations are recovered in the Euclid space endowed with the parallel transport by translation. These issues are treated in detail in the cited references.

In conclusion the answer to the question is:

Balance of linear momentum (which is part of Euler's formulation of the law of Dynamics) is the outcome of choosing in the Euclid space the standard connection associated with the parallel transport by translation. It is a special form of a general expression which doesn't depend on the choice of a connection.

Thank you for your detailed and informative explanation. I agree that Euler-Lagrange equations coming from an action principle are all written in terms of partial derivatives. I think one thing one should consider carefully is that the Lagrangian density in elasticity explicitly depends on the metrics. This then leads to writing the EL equations in the equivalent and well-known form using the Levi-Civita connection (divergence of stress...). The following reference is relevant.

Let me also add one more point. Having a volume form divergence is defined using the Lie derivative. Having a connection one can define the compatible volume form (in the obvious way). It turns out that the standard Riemannian volume form is the compatible volume form of the Levi-Civita connection.

I have a few general questions. Is differential geometry being applied to the study of fluid mechanics at all? Are there people and or universities conducting research in this area? Is it at all effective to apply differential geometry to fluid mechanics?